Copyright  1998 by the Genetics Society of America
Correlation Between the Substitution Rate and Rate Variation
Among Sites in Protein Evolution
Jianzhi Zhang and Xun Gu
Institute of Molecular Evolutionary Genetics and Department of Biology, Pennsylvania State University, University Park, Pennsylvania 16802
Manuscript received December 10, 1997
Accepted for publication April 7, 1998
ABSTRACT
It is well known that the rate of amino acid substitution varies among different proteins and among
different sites of a protein. It is, however, unclear whether the extent of rate variation among sites of a
protein and the mean substitution rate of the protein are correlated. We used two approaches to analyze
orthologous protein sequences of 51 nuclear genes of vertebrates and 13 mitochondrial genes of mammals.
In the first approach, no assumptions of the distribution of the rate variation among sites were made,
and in the second approach, the gamma distribution was assumed. Through both approaches, we found
a negative correlation between the extent of among-site rate variation and the average substitution rate
of a protein. That is, slowly evolving proteins tend to have a high level of rate variation among sites, and
vice versa. We found this observation consistent with a simple model of the neutral theory where most
sites are either invariable or neutral. We conclude that the correlation is a general feature of protein
evolution and discuss its implications in statistical tests of positive Darwinian selection and molecular time
estimation of deep divergences.
T
HE rate of amino acid substitution varies considerably among different proteins as well as different
sites of a protein (e.g., Kimura 1983; Nei 1987; Li 1997).
It is interesting to study the relationship between these
two types of rate variations because they result from
the variation of functional constraints at different levels
(proteins or amino acid residues) according to the neutral theory of molecular evolution (Kimura and Ohta
1974; Kimura 1983). In a recent study of vertebrate
mitochondrial genes, Kumar (1996) observed that the
extent of rate variation among nucleotide sites of a
gene and the average substitution rate of the gene are
negatively correlated. That is, fast evolving genes have
low degrees of among-site rate variation, whereas slowly
evolving genes have high degrees of among-site rate
variation. However, the cause of the correlation was
unclear at the time. Furthermore, the result was obtained based on the strong assumption that the amongsite rate variation follows a gamma distribution that may
or may not hold in all the genes. To examine whether
such correlation is a general feature of molecular evolution, a large-scale analysis of nuclear and mitochondrial
genes with an approach that is free from the assumption
of the gamma distribution seems to be necessary. In this
article, we analyzed protein sequences for 51 nuclear
genes of vertebrates and 13 mitochondrial genes of
mammals and found that the correlation holds for both
nuclear and mitochondrial genes. We also explain the
underlying cause of the correlation and discuss its implications.
DATA AND METHODS
Characterization of among-site rate variation: Let us
assume that the substitution rate (r) at an amino acid
site of a protein sequence is an identically and independently distributed random number following an unknown distribution g (r). Let k be the number of substitutions at a given site during a period of evolutionary time
T. Assuming that amino acid substitutions follow the
Poisson process when the rate is given, k follows the
distribution:
f (k) 5
#
∞
0
(rT)ke2rT
g (r)dr.
k!
(1)
The expectation of k, E(k), can be obtained by
E(k) 5
5
5
∞
o
k50
∞
o
k50
#
3 k#
∞
0
∞ ∞
0
5
[kf(k)]
o0 3k
(rT)ke2rT
g (r)dr
k!
4
(rT)ke2rT
g(r)dr
k!
4
∞
# rTg(r)dr
0
5 mT,
Corresponding author: Jianzhi Zhang, 322 Mueller Laboratory, Institute of Molecular Evolutionary Genetics, Pennsylvania State University, University Park, PA 16802. E-mail: jxz128@psu.edu
Genetics 149: 1615–1625 ( July 1998)
(2)
where m 5 E(r) is the expectation of r, or mean substitution rate. The variance of k, Var(k), can be obtained by
1616
J. Zhang and X. Gu
Var(k) 5
5
5
∞
o
k50
∞
∞
5
0
o 3k 2 #
k 0
#
∞ ∞
#
∞
o0 3k 2
0
5
[k 2 f(k)] 2 [E (k)]2
(rT)ke2rT
g (r)dr 2 (mT )2
k!
4
(rT )ke2rT
g (r)dr 2 (mT )2
k!
4
(r 2T 2 1 rT )g(r)dr 2 (mT )2
0
5 var (r)T 2 1 (mT )2 1 mT 2 (mT)2
5 Var(r)T 2 1 mT,
(3)
where Var(r) is the variance of r. Therefore, the mean
and variance of r can be obtained from
m 5 E(r) 5 E (k)/T
(4)
Var(r) 5 [Var (k) 2 E(k)]/T 2 .
(5)
and
The degree of among-site rate variation is better measured by the coefficient of variation, which is obtained
from
CV(r) 5
√Var(r)
E(r)
5
√Var(k) 2 E(k)
E (k)
.
(6)
It is interesting to see from Equations 4 and 6 that if
we can estimate the mean and variance of k, we will be
able to study the mean substitution rate and the extent
of among-site rate variation without knowing the underlying rate distribution g(r) (Gu and Li 1998).
Nevertheless, the gamma distribution has been predominantly used in recent years to characterize the
among-site rate variation of proteins (e.g., Uzzell and
Corbin 1971; Golding 1983; Tamura and Nei 1993;
Yang 1994; Gu et al. 1995; Gu and Zhang 1997). That
is, the substitution rate (r) at a site is assumed to follow
the gamma distribution with the probability density
function
g (r) 5
(a/m)a a21 2(a/m )r
r e
,
C(a)
(7)
where C( ) is the gamma function. There are two parameters in Equation 7: the mean rate m 5 E(r) and
the shape parameter a. The extent of among-site rate
variation can be measured by a. Small values of a indicate a high degree of rate variation among sites, whereas
large values indicate a low degree of variation. In fact,
under the gamma model, CV(r) 5 1/√a. Some examples of the gamma distribution with different values of
m and a are given in Figure 1.
Genes used: We randomly chose 51 nuclear genes
(see Table 1) for which the orthologous sequences from
the human (Homo sapiens), mouse (Mus musculus), chicken
(Gallus gallus), and African clawed frog (Xenopus laevis)
are available in the GenBank. All the protein sequences
Figure 1.—The probability density functions of gamma distributions. (A) The mean (m) is fixed at 1 and the shape
parameter (a) is presented next to each curve. (B) a is fixed
at 2 and m of each curve is presented.
were downloaded from the HOVERGEN database (Duret
et al. 1994). Hedges et al. (1996) examined the orthology
of most of the genes for their purpose of estimating
the divergence time of primates and rodents. For the
remaining genes, we examined the orthology by phylogenetic analysis as described in Hedges et al. (1996).
All the sequences used were longer than 100 amino
acids. We chose the four species because (1) estimation
of the gamma shape parameter a requires at least three
sequences, (2) there are many gene sequences available
for these species, (3) the phylogeny of the species is
known, and (4) the divergence among these species is
moderate for most genes so that pairwise distances and
substitution rates can be accurately estimated. As shown
in Equation 4, to study the mean rate m, we have to
know the total amount of evolutionary time T for a tree.
But if we use the same species for all the genes, T will
be the same so that we can infer properties of m from
those of mT without knowing the actual value of T.
Rate Variation Among Sites
We also studied 13 mitochondrial genes of 9 mammalian species, including the mouse (GenBank accession
no. J01420), rat (Rattus norvegicus; X14848), human
(D38112), gibbon (Hylobates lar; X97707), cow (Bos taurus; V00654), whale (Balaenoptera physalus; X61145), opossum (Didelphis virginiana; Z29573), wallaroo (Macropus robustus ; Y10524), and platypus (Ornithorhynchus anatinus;
X83427). In this case only mammalian species were used
because mammalian mitochondrial genes evolve quickly
(Brown et al. 1979) and use of distantly related species
may increase estimation errors. The unrooted tree of
these 9 species has been well established (e.g., Janke et
al. 1997).
Sequence analysis: For the orthologous sequences of
each gene, we used the known species tree and estimated the number of substitutions (k) along the tree
at each amino acid site by using Equation 7 of Gu and
Zhang’s (1997) method. We then estimated the mean
and variance of k. Estimates of mT and CV(r) were then
obtained by using Equations 4 and 6, and they were
denoted as D and CV, respectively.
We also estimated the gamma shape parameter (a)
for the protein sequences of each gene by using Gu
and Zhang’s (1997) method, which has been shown to
be as accurate as, but much faster than, the likelihood
method. Actually, the likelihood method (Yang 1994)
and the Gu and Zhang’s (1997) method gave essentially
the same result for the data sets used in this article (data
not shown). In Gu and Zhang’s (1997) method, the k
value for each site is first estimated. Because k follows
the negative binomial distribution when r follows the
gamma distribution, the gamma shape parameter a is
estimated by maximizing the following likelihood function derived from the negative binomial distribution,
L(a) 5
p3
i 1
n
5
C(a 1 ki) D
ki !C(a) D 1 a
1
2 1D 1a a2 4 ,
ki
a
(8)
where ki is the number of substitutions at site i, n is the
number of amino acids of the sequence, and D 5 mT
is the mean of ki over all sites, as mentioned earlier.
Under the gamma model, the unbiased pairwise distances among the orthologous sequences were estimated by using the formula (Nei et al. 1976; Ota and
Nei 1994),
d 5 a[(1 2 p)21/ a 2 1] ,
(9)
where d is the gamma distance between two protein
sequences, p is the proportional difference, and a is the
estimated gamma shape parameter.
For each gene, by a likelihood ratio test, we examined
whether the null hypothesis that all sites have the same
substitution rate can be rejected and whether the alternative hypothesis of the gamma distribution is favored.
We computed
l 5 2ln[L(a)/L (∞)],
(10)
1617
where a is the estimated gamma shape parameter and
L (∞) is the likelihood of the null hypothesis. Because
the null hypothesis is a special case of the alternative
hypothesis with a being infinity, l is approximately x2 distributed with the number of degrees of freedom being 1.
RESULTS
Nuclear genes: Table 1 shows the estimates for the
mean substitution rate and the extent of among-site
rate variation for the 51 nuclear genes examined. The
coefficient of variation (CV) of the substitution rate
among sites varies from 0.43 to 2.02 with the mean of
1.09 and the median of 1.00 (Figure 2). Figure 3 shows
the relationships between CV and D (average number
of substitutions per site along the whole tree). We fitted
the data by a linear regression of CV and lnD. The
coefficient of correlation (R) is 20.95, which is significantly different from 0 (P , 0.0001). If we fit the data
by a linear regression of CV and D, R becomes 20.91
(P , 0.0001). Because D 5 mT and T (total amount of
evolutionary time of the tree) is identical for all the
genes, the correlation coefficient between CV and D
(or ln D) is the same as that between CV and m (or lnm).
These results therefore show that the mean substitution
rate and the extent of among-site rate variation of a
protein are negatively correlated. This observation is
independent of any assumption about the underlying
distribution of the substitution rate g(r).
We then examined the correlation under the assumption of a gamma distribution for the rate variation
among sites. The gamma shape parameter a of the 51
nuclear genes varies from 0.17 to 3.45 with the median
of 0.71 (Table 1 and Figure 4). About two-thirds of
the genes have a values ,1, and, therefore, their rate
distributions are L-shaped (see Figure 1A). Only 12%
of the genes have a values .2. Likelihood ratio tests
showed that the null hypothesis of no rate variation
among sites can be rejected at the 1% significance level
for 48 of the 51 genes and can be rejected at the 5% level
for the remaining three genes. Therefore, the alternative
hypothesis of the gamma distribution is favored. A common feature of the three genes mentioned above is that
they are all short (around 150 amino acids), suggesting
that the test might not be powerful enough for these
genes. The gamma distance d (number of amino acid
substitutions per site) between the mouse and chicken
varies from 0.057 to 0.703 for the 51 genes (Table 1),
which corresponds to the rate of about 0.1 to 1.2 amino
acid substitutions per site per billion (109) years if we
assume that mammals and birds diverged about 300 mya,
which is now generally accepted (Benton 1993; Hedges
et al. 1996).
Figure 5 shows the relationship of a and d (between
the mouse and chicken) for the 51 nuclear genes. The
R value for a and d is 0.68, which is significantly different from 0 (P , 0.0001). Linear regression of a and lnd
does not change the R value very much. Because only
1618
J. Zhang and X. Gu
TABLE 1
Substitution rates and among-site rate variations of 51 nuclear genes of vertebrates
Genes
Acetylcholine receptor a subunit
Acetylcholine receptor d subunit
Acetylcholine receptor g subunit
a-Globin
Annexin II
B-myb
Bcl2-Ig fusion gene
Beta 2 thyroid hormone receptor
C-kit proto-oncogene
C-myb
C-yes protein
CD18-integrin b-2
CDC2
Complement component 3
Connexin 43
Cyclin A
E-cadherin
Erythroblast virus oncogene homolog 2
Estrogen receptor
Focal adhesion
Follistatin
GATA-3-transcription factor
Hedgehog
Histon H2A.X
Lactate dehydrogenase-A
Lamin B1
Midkine protein
MOS proto-oncogene
Muscle pyruvate kinase
Myogenic factor Myf-5
N-cadherin
N-myc
Na1/K1 ATPase b-1-subunit
Na1/K1 ATPase b-2-subunit
NAD(1) ADP-ribosyltransferase
Nerve growth factor
Neural cell adhesion molecule
Neurotrophin-3
Nucleolar protein N038
Nucleolin
Ornithine decarboxylase
P53 cellular tumor antigen
Prohormone convertase
Rhodopsin
S6 II kinase
SPARC
TGF b-2
TGF b-3
Transcription factor Eryf1
Tyrosine phosphatase
Vimentin
a
Accession numbersa
Amino acids
CV
Db
a
pc
dd
P02708; X03986; X07330; X17244
X55019; K02582; K02903; X07069
X01715; X03818; K02904; X07068
V00493; V00714; J00855; X14259
M14043; M14044; X53334; M60768
X13293; X70472; X67505; M75870
X06487; M16506; Z11961; X82461
X04707; U15542; M65207; M35359
X06182; Y00864; D13225; M80798
X52125; M12848; M14129; L22741
M15990; X67677; X13207; X14377
X64072; X14951; X71768; S49364
X05360; M38724; X16881; M60680
K02765; K02782; U16848; U19253
X52947; X62836; M29003; X17243
X68303; Z26580; X72892; X53745
Z13009; X06115; M22180; U04708
J04102; J04103; X07202; X52635
X99101; M38651; X03805; A40907
L13616; M95408; M86656; U11078
P19883; Z29532; X87609; P31515
X55122; X55123; X56931; M76565
L38518; X76290; L28099; L35248
X14850; X58069; V00413; M21287
X02152; U13687; X53828; U07179
L37737; X16705; X16878; X06344
X55110; M19662; X76482; U06048
J00119; J00620; M19412; X13311
X56494; D38379; J00903; U03878
X14894; X56182; X73250; X56738
X54315; M31131; X07277; X57675
Y00664; X03919; D90071; X58670
M25160; X16646; J02787; U17061
M81181; X16645; L13208; M37788
P09874; P11103; P26446; P31669
X52599; M14805; X04003; X55716
X16841; Y00051; M15860; M25696
M37763; M53257; M83378; Z30090
M28699; M33212; X17200; X56039
M60858; X07699; X17199; X72957
P11926; P00680; X64710; X56316
X02469; X00741; X13057; M36962
X17094; X54056; Z68093; M80471
U49742; M55171; D00702; S79840
L07597; M28489; M28488; M20187
J03040; X04017; L24906; X62483
M19154; X57413; M31154; X51817
X14149; M32745; X58071; X51817
X17254; X15763; M26209; M76566
X53364; M36033; L22437; U09135
Z19554; X51438; M15850; X16843
456
508
502
142
339
624
161
368
909
615
535
790
295
1493
378
365
856
464
580
1051
315
430
408
139
330
577
140
328
527
255
899
404
301
273
991
230
713
228
282
594
448
343
590
348
718
296
409
389
268
791
451
1.28
0.83
0.70
0.63
1.21
0.85
0.50
1.98
0.44
1.01
1.96
1.30
1.50
0.47
1.69
0.75
0.70
1.00
0.97
2.00
1.51
1.64
1.29
2.02
1.03
1.00
0.68
0.64
1.42
0.90
1.37
0.94
0.90
0.62
1.00
0.80
1.10
0.43
0.79
0.60
1.22
0.53
1.34
1.28
1.80
1.27
1.70
0.73
0.62
1.50
1.35
0.51
0.82
0.81
1.02
0.37
0.94
1.54
0.12
1.77
0.72
0.20
0.42
0.26
1.20
0.27
1.28
1.09
0.73
0.60
0.20
0.33
0.39
0.68
0.20
0.65
0.65
0.92
1.12
0.28
0.88
0.46
0.82
0.78
1.11
0.55
0.90
0.65
1.51
0.86
0.95
0.43
1.24
0.45
0.32
0.32
0.50
0.31
0.99
1.23
0.51
0.43
0.41
1.05
1.67
1.93
0.55
0.93
2.59
0.21
3.45
0.68
0.20
0.43
0.31
3.29
0.24
1.14
1.31
0.70
0.81
0.17
0.33
0.24
0.36
0.19
0.71
0.82
1.63
1.69
0.38
0.79
0.38
0.77
0.86
2.08
0.77
0.99
0.59
2.45
1.26
2.23
0.47
1.81
0.40
0.44
0.22
0.42
0.22
1.04
1.95
0.27
0.38
0.158
0.315
0.323
0.319
0.103
0.268
0.236
0.049
0.343
0.184
0.092
0.153
0.092
0.444
0.069
0.307
0.379
0.226
0.203
0.087
0.108
0.072
0.164
0.085
0.146
0.199
0.371
0.384
0.140
0.314
0.137
0.312
0.286
0.487
0.218
0.357
0.138
0.110
0.301
0.370
0.174
0.436
0.137
0.135
0.079
0.151
0.117
0.123
0.452
0.173
0.111
0.214
0.455
0.439
0.425
0.120
0.370
0.284
0.057
0.447
0.237
0.124
0.203
0.112
0.643
0.083
0.432
0.573
0.310
0.263
0.119
0.136
0.088
0.232
0.114
0.176
0.255
0.537
0.561
0.186
0.482
0.179
0.482
0.412
0.787
0.290
0.555
0.168
0.119
0.415
0.514
0.236
0.672
0.179
0.172
0.101
0.199
0.168
0.140
0.703
0.275
0.138
The Gen Bank/EMBL/SwissProt accession numbers are presented in the order of human, mouse, chicken, and frog sequences.
Average number of substitutions per site along the tree.
c
Proportional differences between the mouse and chicken proteins.
d
Gamma distances between the mouse and chicken proteins.
b
Rate Variation Among Sites
Figure 2.—Distributions of the CV values of 51 nuclear
(solid histograms) and 13 mitochondrial (open histograms)
genes.
orthologous sequences were used, d is proportional to
the average substitution rate m (d 5 2mt, where t is
the time since the divergence of birds and mammals).
Therefore, the correlation coefficient between a and m
is also 0.68. Although only the distances between the
mouse and chicken are presented in Figure 5, use of
pairwise distances among other species gave similar results. Large estimates of a (e.g., a . 2) are known to
have substantial sampling errors (Gu and Zhang 1997),
and they should have lower weights than other data
points in the regression. A simple practice is to give
Figure 3.—Correlation between the extent of among-site
rate variation CV and the mean substitution rate D (5mT ) in
51 nuclear genes. R is the coefficient of correlation.
1619
Figure 4.—Distributions of the a values of 51 nuclear (solid
histograms) and 13 mitochondrial (open histograms) genes.
zero weights to them. The correlation coefficient for the
remaining 45 genes with a , 2 is now 0.86 (P , 0.0001).
Influence of estimation biases: The above analyses
showed that the estimated mean substitution rate and
the extent of rate variation among sites are highly correlated. However, to prove that our observations reflect
the true properties of protein evolution, we have to
Figure 5.—Correlation between the gamma shape parameter a and the distance d between mouse and chicken proteins.
The dashed line is the regression for 51 genes (both solid
and open circles), whereas the solid line is the regression for
the 45 genes (solid circles) excluding the 6 genes (open circles) whose a values are estimated to be greater than 2. The
regression equations presented at the bottom of the figure
are for the 51 genes, and the equations at the top are for the
45 genes.
1620
J. Zhang and X. Gu
show that the correlation is not an artifact due to some
biases of the estimation methods we have used. In the
estimation of CV and D, no assumptions of the distribution g(r) were made. It is, therefore, very difficult to
examine the estimation biases quantitatively. In the following paragraphs, we studied possible biases in the
estimation of a and d, where g(r) was assumed to be
gamma.
A computer simulation study (J. Zhang, unpublished
results) demonstrated that when the number of sequences used is small (e.g., four), estimation of a is
subject to asymptotic bias: a is likely to be underestimated for closely related sequences, but overestimated
for highly diverged sequences. This is because when the
divergence level is low, there is not enough time for
substitutions to occur at potentially variable sites, so they
are regarded as invariant sites and a is underestimated.
When the divergence level is very high, multiple hits
cannot be fully corrected. As a result, the numbers of
substitutions are underestimated at very fast evolving
sites and a is subsequently overestimated. The estimation bias decreases when the number of sequences used
increases (Gu and Zhang 1997; J. Zhang, unpublished
results). Because the estimation bias of a is likely to
create a positive correlation between a and d (or m), it
is worthwhile to examine the influence of the estimation
bias on the present study quantitatively by using computer simulation.
The tree of the human, mouse, chicken, and frog
(Figure 6) was used as a model tree in the simulation.
The branch lengths of the tree were proportional to
the divergence times of the species, that is, 100 million
yr between the human and mouse (Hedges et al. 1996),
300 million yr between mammals and birds, and 350
million yr between amniotes and frogs (Ahlberg and
Milner 1994). In the computer simulation, we simulated the evolution of 45 independent genes. For each
gene, we randomly chose an a value from a uniform
distribution in the range of 0.2 to 2 as the gamma shape
parameter of the gene, and a p value from a uniform
Figure 6.—The model tree used in the simulation for nuclear genes.
distribution in the range of 0.05 to 0.45 as the proportional difference between the mouse and chicken proteins. These ranges represented the observed variations
in the 45 genes, excluding the 6 genes with estimated
a values .2. The sequence length used was 450 amino
acids, because it is approximately the average length of
the proteins that we have examined. The simulation
procedure was similar to that described in Zhang and
Nei (1997), except that the substitution model used was
gamma 1 Poisson. That is, for each site, the relative
substitution rate (rR ) was randomly generated from a
gamma distribution with the expectation of 1 and the
shape parameter a, and the amino acid substitution
followed the Poisson process. The tree branch lengths
(expected numbers of substitutions per site) were computed for each site separately. For example, the expected distance between the mouse and chicken is dM2 C,
computed from Equation 9 by using the a and p values
that were randomly chosen for this gene, and the average substitution rate for the protein is then m 5 dM2C/
(2 3 300) substitutions per site per mya, where 300
stands for 300 million yr. The substitution rate at a given
site is therefore equal to r 5 rRm, where rR is the relative
rate at the site. The length of the branch from human
to the common ancestor of human and mouse is then
100r substitutions for the site, where 100 stands for 100
million yr. Other branch lengths can be computed similarly. In the simulation, after the four protein sequences
at the exterior nodes of the tree were generated, the
gamma shape parameter a was estimated by the method
of Gu and Zhang (1997), and the gamma distance (d)
between the simulated mouse and chicken sequences
was computed by using Equation 9 with the estimated
a and p. The simulation results with the estimated a
.2 were abandoned. When 45 independent genes were
simulated and 45 couples of a (,2) and d were obtained, the correlation coefficient R was computed. We
conducted 2500 replications of such computer simulations, and the distribution of the 2500 R values is presented in Figure 7.
It is seen from Figure 7 that the estimated a and d
are positively correlated, though the true a and d have
no correlation at all. The correlation coefficient of the
estimated a and d is significantly different from 0 because all 2500 simulated R values are positive. The average correlation coefficient is 0.65. However, only 0.32%
of the 2500 simulated R values are greater than 0.86,
which is the observed R value for the 45 nuclear genes
examined. In other words, the observed R value of 0.86
is significantly greater than 0 at the 0.32% significance
level, even when the estimation bias of a is considered.
Therefore, the observed positive correlation between a
and d of the nuclear proteins seems to have underlying
causes other than the estimation bias.
In the above simulation, a and p were randomly chosen from a uniform distribution, respectively. But in
reality, neither of them follow a uniform distribution
Rate Variation Among Sites
Figure 7.—The distribution of 2500 R values of the estimates of a and m in the simulation. There is no correlation
between the true values of a and m. The observed R from the
45 nuclear genes is marked.
(see Figure 4). It is expected, and is confirmed in a
small-scale simulation, that the R values obtained from
the simulation will be smaller when the true distributions of the a and p are used. Therefore, the probability
of observing an R value of 0.86 or larger, given that
there is no correlation between a and d, may have been
overestimated in the simulation. In other words, the
effect of the estimation bias may be exaggerated in the
simulation and the confidence level of 0.32% is a conservative estimate. The assumptions of the Poisson model
of amino acid substitution for protein evolution does
not seem to have affected our simulation result very
much. Because only orthologous proteins are used in
the analysis, the evolutionary rate of a protein is expected to be more or less constant among lineages unless there is variation of the mutation rate due to the
generation time effect (Ohta 1995; Li et al. 1996),
which, however, acts on all the genes to the same extent
and does not affect the correlation between CV (or a)
and D (or d).
Mitochondrial genes: The estimated CV values of the
13 mitochondrial genes are between 0.66 and 2.05, with
the mean of 1.14 and the median of 1.10 (Table 2 and
Figure 2). Again, we found that CV and D are highly
correlated. The R value is 20.98 (P , 0.0001) for the
linear regression between CV and ln D (Figure 8). If we
fit the data by a linear regression of CV and D, R becomes
20.92 (P , 0.0001). These results show that the mean
substitution rate and the extent of among-site rate variation are negatively correlated for the mitochondrial genes
as well.
The estimated a values for the mitochondrial genes
vary from 0.19 to 1.68 with the median of 0.56 (Table
1621
Figure 8.—Correlation between CV and D in 13 mammalian mitochondrial genes.
2; Figure 4). Similar to the nuclear genes, about twothirds of the mitochondrial genes have a values ,1.
The distance d between the human and mouse varies
from 0.20 to 1.15 (Table 2), corresponding to the rate
of about 1.0 to 5.8 amino acid substitutions per site per
billion years under the assumption that the human and
mouse diverged 100 mya. It is seen that mammalian
mitochondrial genes evolve about 5 to 10 times faster
than nuclear genes, as generally believed (Brown et al.
1979).
Figure 9 shows the relationship of a and d (of the
TABLE 2
Substitution rates and among-site variations of 13
mitochrondrial genes of mammals
Genes
Amino acids
CV
Da
a
pb
dc
Atp6
Atp8
Co1
Co2
Co3
Cytb
Nd1
Nd2
Nd3
Nd4
Nd4l
Nd5
Nd6
226
60
512
222
261
379
315
345
113
457
97
593
160
1.10
0.69
2.05
1.43
1.74
1.27
1.12
0.73
1.12
1.01
0.92
0.95
0.66
1.26
2.82
0.37
0.84
0.66
0.99
1.17
2.33
1.64
1.48
1.71
1.94
2.39
0.51
1.38
0.19
0.45
0.24
0.44
0.56
1.31
0.42
0.64
1.22
0.70
1.68
0.257
0.567
0.092
0.226
0.134
0.222
0.225
0.423
0.336
0.335
0.330
0.359
0.450
0.403
1.151
0.126
0.445
0.197
0.338
0.323
0.683
0.693
0.571
0.474
0.621
0.718
a
Average number of substitutions per site along the whole
tree.
b
Proportional differences between the human and mouse
proteins.
c
Gamma distances between the human and mouse proteins.
1622
J. Zhang and X. Gu
Figure 9.—Correlation between a and the distance d of
human and mouse in 13 mammalian mitochondrial genes.
human and mouse) for the mitochondrial genes. The
correlation coefficient between a and d (or m) is 0.72
(P , 0.01). Because we have used nine species in the
estimation of a, and the estimation bias of a is expected
to be quite small when the number of sequences used
is no less than eight (Gu and Zhang 1997; J. Zhang,
unpublished results), the observed correlation between
a and d is unlikely to have been affected by the estimation bias of a. This result was also confirmed by a smallscale computer simulation similar to that performed
for the nuclear genes (data not shown). As mentioned
before, Kumar (1996) found that the correlation coefficient between a and d for mitochondrial genes is 0.82
by using 1st and 2nd codon positions of the nucleotide
sequences. This value is quite close to what we have
obtained (0.72), although we used different species and
different methods of a estimation. This suggests that
the correlation is robust. As in the case of the nuclear
genes, the hypothesis of no rate variation among sites
was rejected for all 13 mitochondrial genes (P , 0.01).
DISCUSSION
Causes of the negative correlation between the mean
rate and the extent of among-site rate variation: Through
the analysis of 64 genes in total, we have shown that the
mean substitution rate and the extent of among-site rate
variation of a protein are negatively correlated. We have
also excluded the possibility that the correlation is
caused by estimation biases or is dependent on the assumption of the gamma distribution of the rate variation
among sites. Because the 51 nuclear genes we examined
are scattered all over the genome and they encode proteins with a variety of functions, it is unlikely that the
correlation is limited to particular chromosomal regions
or is restricted to particular groups of proteins. Rather,
the correlation is likely to be a general feature of molecular evolution because both nuclear and mitochondrial
genes show similar patterns. Nevertheless, all the data
we used are from vertebrates, and our conclusion needs
further support from genes of other organisms.
The observation of negative correlation can be explained under the neutral theory of molecular evolution
(Kimura 1983; see also Ohta 1992), which asserts that
protein sequence evolution is under purifying selection
and random genetic drift, and that the rate variation is
due to the difference in functional constraints among
amino acid sites of a protein and among different proteins. In the following paragraphs, we explain the cause
of the correlation by using some simple models of neutral evolution.
Let n be the mutation rate at an amino acid site,
which is assumed to be constant for all sites of all proteins. Then the substitution rate is 0 for an invariant
site, between 0 and n for a variable but functionally
constrained site, and equal to n for a neutral site. Let
us first consider a simple model in which the functional
constraints at all sites of a protein are equal. Then the
mean rate m and CV have no correlation because CV is
always 0, and m can be any value from 0 to n. Therefore,
this model cannot explain our observation of the correlation between CV and m. Now let us consider another
model, in which there are two types of sites, either invariant or neutral. Then the average substitution rate of
the protein becomes
m 5 nf,
(11)
where f is the proportion of neutral sites in the protein
(Kimura 1983). The variance (Var) and the coefficient
of variation (CV) of the rate among sites can be computed as
Var 5 f(n 2 fn )2 1 (1 2 f )(0 2 f n )2
5 f(1 2 f )n2
(12)
and
CV 5
√Var
m
5
n√f (1 2 f )
1
5
2 1.
fn
f
!
(13)
By using Equations 11 and 13, we obtained
m5
n
.
CV 1 1
2
(14)
Equation 14 shows that, under this model, m is negatively
correlated with CV and is dependent only on CV, when
n is given. Use of more complex models (e.g., Kimura
1979) gives qualitatively similar results. With Equation
14 in mind, we computed the correlation coefficient R
for D and 1/(CV 2 1 1). It was 0.95 (P , 0.0001) and
0.97 (P , 0.0001) for the nuclear and mitochondrial
genes, respectively.
In reality, the pattern of the variation of functional
Rate Variation Among Sites
constraint among sites is somewhere between the above
two extreme models, and the observation of the strong
correlation between m and CV or 1/(CV 2 1 1) suggests
that it is probably much closer to the second model
than to the first. This is biologically understandable
because, for an amino acid sequence to be functional,
there must be some essential sites that are invariant and
some other sites that are from mildly variable to almost
free of any change. This pattern of functional constraints seems to be the requirement for a functional
protein. It is unrealistic that the constraints at all amino
acid sites of a protein are more or less equal unless the
protein is free of any functional constraint (no function). In Equation 14, we can see that v also influences
the relationship of m and CV. There is evidence that
the mutation rate varies among nuclear genes (Wolfe
et al. 1989). Therefore, the correlation between m and
CV for the nuclear genes may have been depressed due
to the variation of mutation rate among genes. This
effect is expected to be small for the 13 mitochondrial
genes because they have similar mutation rates (Nedbal
and Flynn 1998).
The biological meaning of a: The gamma distribution
is predominantly used to fit the rate variation among
sites of proteins. Our results from the likelihood ratio
test indicated that the gamma distribution model fit the
rate variation significantly better than the uniform rate
model. But the reason why rate variation follows the
gamma distribution is unknown and, therefore, the biological meaning of the shape parameter a is not clear.
According to our observation of the strong positive correlation between a and m, a can be interpreted as a
measure of functional conservation of a protein. Small
values of a indicate strong constraint, and large values
indicate weak constraint. It is therefore expected that
fourfold degenerate sites and intron regions of functional genes and pseudogenes have very large a values.
For example, we estimated that the a of the glycerol3-phosphate dehydrogenase (GPDH, EC 1.1.1.8) of 13
Drosophila and related species (Kwiatowski et al.
1997) is about 0.10, but the a of the intron 3 and 5 of
the Gpdh gene is z3.0. Actually, the hypothesis of rate
constancy over sites cannot be rejected for these intron
sequences (P . 0.05, goodness of fit test). However, the
above interpretation of a is valid only when the majority
of amino acid sites are undergoing neutral evolution
(i.e., no positive selection). When strong positive selection is acting on a large number of sites, m tends to be
high and a tends to be small because functionally most
important sites remain invariant, whereas positively selected sites have very high substitution rates. So, in this
case, a does not measure the functional conservation
of a protein. For example, we found that the a value
of the sperm lysin of 20 abalone species (Lee et al. 1995)
is 0.16, whereas the m is about 2 to 3 times the mutation
rate n. Here, n was estimated under the assumption
that the mutation rate per amino acid is 2.25 times the
1623
mutation rate per nucleotide (Nei 1975, p. 225), which
was further assumed to be equal to the rate of synonymous nucleotide substitution. So, if this gene is presented in Figure 5, it will sit at the top left corner, far
from the regression line.
It is intriguing that there are no dots sitting in the
upper left corner in Figure 5 or the upper right corner
in Figure 3. These are the locations where both the
mean rate and the among-site rate variation are high,
and where we expect to find proteins consisting of a
number of conserved sites and a number of directionally
selected sites. The simplest explanation would be that
such genes are not included in our sample of 64 genes.
Because our sample is random and is not small, the
result suggests that those genes that are under strong
positive selection are rare in the genome. Of course,
we cannot distinguish neutral genes from those in which
positive selection operates in a very small number of
sites, because in the latter, the overall relationship between a and m is mainly determined by the sites that
are neutral or under purifying selection. In fact, there
are no statistical methods that can effectively detect
positive selection that only acts on one or several sites
except in the case of convergent and parallel evolution
(Zhang and Kumar 1997). It has to be noted that all
we have studied here is orthologous gene evolution.
Because orthologous genes usually have the same function, it is understandable that we did not find signals
of strong positive selection (Sharp 1997). Strong selection is more likely to operate on newly duplicated genes
during the evolution of novel gene functions (Zhang
et al. 1998).
It is also interesting that no dots were found in the
lower left corner of Figure 3, where both the mean rate
and the rate variation are small. This suggests that even
in the most conserved proteins it is unlikely that all sites
are equally conserved. More probably, there are still a
few nearly neutral sites so that CV becomes high (see
Equation 13).
Other implications: Accelerated evolution in some
evolutionary lineages has been observed in several proteins (e.g., Stewart et al. 1987; Tucker and Lundrigan
1993; Wallis 1993, 1996; Whitfield et al. 1993). If the
mutation rate is constant among lineages, there are two
possible causes of the enhanced substitution rate. One
is relaxation of functional constraint, and the other is
positive selection. When the rate of nonsynonymous
nucleotide substitution is not significantly greater (or
even not greater) than the rate of synonymous substitution, it is often difficult to distinguish between the two
possible causes of the accelerated evolution (e.g., Li and
Gojobori 1983). The relationship between m and a
may provide us some information, because when m is
high, high values of a are expected if functional constraint is reduced, but small values of a are likely if
strong positive selection acts on a large number of sites
(as in the case of the abalone sperm lysin).
1624
J. Zhang and X. Gu
Using the relationship between m and a, let us examine an interesting case of fast evolution. Whitfield et
al. (1993) and Tucker and Lundrigan (1993) reported
that the sex-determining protein SRY evolves very fast
in primates and rodents, and the rates of nonsynonymous and synonymous nucleotide substitution are similar. But because the nonsynonymous rate is not significantly higher than the synonymous rate, these authors
were not able to determine whether the accelerated
evolution is due to positive selection or relaxation of
functional constraints. SRY protein is a transcription
factor that binds to specific DNA sites to regulate the
expression of its downstream genes in the sex determination pathway. SRY consists of N-terminal, HMG, and
C-terminal domains (with 58, 78, and 68 amino acids,
respectively, in the human sequence). The HMG domain is responsible for DNA binding and is very conservative, whereas the other two domains are highly variable, particularly the C domain. The length of the HMG
domain remains the same among placental mammals,
but the lengths of the other two domains vary within
or among mammalian orders. The estimated a values
of the SRY proteins of eight primate species (Whitfield
et al. 1993) are 1.58, 0.24, and 3.73 for the N, HMG,
and C domains, respectively. The large values of a for
the N and C domains, particularly the latter, suggest
that the accelerated evolution in SRY is probably due
to low functional significance out of the HMG domain
rather than adaptation under strong positive selection.
Actually, all clinical mutations in human SRY resulting
in phenotypic sex reversal are found in the HMG domain (Hawkins et al. 1992; Goodfellow and LovellBadge 1993; Werner et al. 1995), except for one case in
which a nonsense mutation happened in the C domain
(Tajima et al. 1994). Pontiggia et al. (1995) found that
SRY proteins from different primate species are very
similar in both DNA binding and bending and predicted
that they should be able to substitute for each other.
Ignoring rate variation among sites causes underestimation of distances between homologous proteins. For
estimating dates of early divergences by the molecular
clock, only slowly evolving proteins can be used. Because
slowly evolving proteins tend to have high degrees of
rate variation among sites, consideration of the rate
variation and use of gamma distances seems to be necessary for such estimations. This has been highlighted in
recent debates on the age of the most recent common
ancestor of present-day organisms (Doolittle et al.
1996; Hasegawa et al. 1996; Miyamoto and Fitch 1996;
Feng et al. 1997; Gu 1997). In this case, neglect of the
rate variation among sites was found to bias the time
estimation at least 0.5–1 billion years. However, it is
worth mentioning that use of the gamma distance enlarges the sampling variance so that many genes are
needed for obtaining reliable estimates of divergence
times.
We are grateful to Sudhir Kumar and Patrick Parker for provid-
ing us aligned protein sequences of the nuclear genes. We thank
Sudhir Kumar, Masatoshi Nei, Tanya Sitnikova, Chung-I Wu,
and two anonymous reviewers for their comments on an earlier version
of this article. This work was supported by the research grants from
the National Institutes of Health and the National Science Foundation
to M. Nei.
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Communicating editor: C.-I Wu